| L(s) = 1 | − 6.70·5-s − 11.6i·7-s + 15.5i·11-s + 8·13-s − 26.8·17-s − 23.2i·19-s − 31.1i·23-s + 20.0·25-s − 13.4·29-s + 11.6i·31-s + 77.9i·35-s − 2·37-s + 40.2·41-s − 23.2i·43-s − 31.1i·47-s + ⋯ |
| L(s) = 1 | − 1.34·5-s − 1.65i·7-s + 1.41i·11-s + 0.615·13-s − 1.57·17-s − 1.22i·19-s − 1.35i·23-s + 0.800·25-s − 0.462·29-s + 0.374i·31-s + 2.22i·35-s − 0.0540·37-s + 0.981·41-s − 0.540i·43-s − 0.663i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4179191076\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4179191076\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 6.70T + 25T^{2} \) |
| 7 | \( 1 + 11.6iT - 49T^{2} \) |
| 11 | \( 1 - 15.5iT - 121T^{2} \) |
| 13 | \( 1 - 8T + 169T^{2} \) |
| 17 | \( 1 + 26.8T + 289T^{2} \) |
| 19 | \( 1 + 23.2iT - 361T^{2} \) |
| 23 | \( 1 + 31.1iT - 529T^{2} \) |
| 29 | \( 1 + 13.4T + 841T^{2} \) |
| 31 | \( 1 - 11.6iT - 961T^{2} \) |
| 37 | \( 1 + 2T + 1.36e3T^{2} \) |
| 41 | \( 1 - 40.2T + 1.68e3T^{2} \) |
| 43 | \( 1 + 23.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 31.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 20.1T + 2.80e3T^{2} \) |
| 59 | \( 1 - 62.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 104T + 3.72e3T^{2} \) |
| 67 | \( 1 - 69.7iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 62.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 61T + 5.32e3T^{2} \) |
| 79 | \( 1 - 92.9iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 77.9iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 147.T + 7.92e3T^{2} \) |
| 97 | \( 1 - 103T + 9.40e3T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.236975536103705133254321007710, −8.475795878092344950539511290737, −7.55161293841816779460186639690, −7.07350572983151701961871963033, −6.54019824833956078184268976995, −4.75274042403441174739100875855, −4.34808947721387432917751022335, −3.74049585135861123837115209434, −2.37215072993284779222685424203, −0.858315317033548896129516147632,
0.14526050983063020521515213326, 1.80996296196706377800845717985, 3.09699210620809449499052393467, 3.71574570850063050314310447410, 4.79338074044487663673143746573, 5.92553645230919867364688267683, 6.22168114863865587815448351150, 7.68646800242655541279378763898, 8.125056757034416243095272316366, 8.924164624426313496578129764243
